NOTE. The reason of the rule is obvious. By taking away the righthand figure, each of the other figures is brought one place nearer to units, and its value is only one tenth as great as before (Art. 15), and .. the whole is divided by 10. For like reasons, cutting off two figures divides by 100; cutting off three figures divides by 1000, etc.

114. Divide 402763 by 10. 115. Divide 76943 by 100.

116. Divide 98765423 by 100000.

Ans. 769 and 43 Rem.

Ans. 987 and 65423 Rem.

117. Divide 3078654321 by 100000000.

82. To divide by 20, 50, 700, or any similar number: RULE. Cut off as many figures from the right of the dividend as there are ciphers at the right of the significant figures of the divisor, and then divide the remaining figures of the dividend by the significant figures of the divisor.

NOTE 1. This is on the principle of dividing by the factors of the divisor; .. the true remainder will be found by the rule in Art. 80.

118. Divide 74689 by 8000.

OPERATION.

8) 74.689

Quotient,

9

...2 Rem.

Ans. 9 and 2689 Rem.

We divide by 1000 by cutting off 689, which gives 74 for a quotient and 689 for a remainder; then divide 74 by 8, and

obtain the quotient, 9, and remainder, 2. This remainder, 2, is 2000, which, increased by 689, gives 2689 for the true remainder (Art. 80).

NOTE 2. It will be observed that the true remainder, in all examples fike the 118th, is obtained by annexing the 1st to the 2d remainder.

119. Divide 67475 by 2400.

120. Divide 74689 by 4200. 121. Divide 276987 by 3300. 122. 769842 - 45000 =? 123. 999999933300 =? 124. 8040708040000 =? 125. 987654321 ÷ 90900 =?

Ans. 17 and 3289 Rem.

Ans. 17 and 4842 Rem.

82. Rule for dividing by 20? By

81. Reason of rule for dividing by 10? 500? Reason? How is the true remainder found?

GENERAL PRINCIPLES OF DIVISION.

83. The value of a quotient depends upon the relative values of the divisor and dividend, and not upon their absolute values, as will be seen by the following propositions.

(a) If the divisor remains unaltered, multiplying the dividend by any number is, in effect, multiplying the quotient by the same number; thus,

i. e. multiplying the dividend by 4 multiplies the quotient by 4. (b) Dividing the dividend by any number is dividing the quotient by the same number; thus,

i. e. dividing the dividend by 3 divides the quotient by 3.

(c) Multiplying the divisor divides the quotient; thus,

i. e. multiplying the divisor by 3 divides the quotient by 3. (d) Dividing the divisor multiplies the quotient; thus,

i. e. dividing the divisor by 5 mult plies the quotient by 5.

83. Does the size of the quotient depend upon the absolute size of divisor and dividend? Upon what does it depend? What is the first proposition? Second! Third? Fourth?

(e) It follows, from (a) and (b), that the greater the dividend, the greater is the quotient; and the less the dividend, the less the quotient.

(f) Also, from (c) and (d), that the greater the divisor, the less is the quotient; and the less the divisor, the greater the quotient.

84. From the illustrations in Art. 83 we see that any change in the dividend causes a SIMILAR change in the quotient, and that any change in the divisor causes an OPPOSITE change in the quotient. Hence,

(a) Multiplying both dividend and divisor by the same number does not affect the quotient; thus,

12÷34

2 2

246 4, Quotient unchanged.

(b) Dividing both dividend and divisor by the same number does not affect the quotient; thus,

20 5)20 4

10=2

5)10

22, Quotient unchanged.

(c) It follows from (a) and (b), that the operations of multiplying and dividing by the same number cancel (i. e. destroy) each other; e. g.,

If a number be multiplied by any number, and the product be divided by the multiplier, the quotient will be the multiplicand; thus,

8 X 756, and 56 78, the multiplicand.

÷

Also, if a number be divided by any number, and the quotient be multiplied by the divisor, the product will be the dividend; thus,

15 3

5, and 5 × 3 = 15, the dividend.

83. What follows from (a) and (b)? From (c) and (d)? 84. Any change in the dividend, how does it affect the quotient? Any change in the divisor, how? First inference? Second? Third? Illustrate.

85. These general principles may be more briefly stated as follows:

1st. Multiplying the dividend multiplies the quotient; and dividing the dividend divides the quotient (Art. 83, a and b).

2d. Multiplying the divisor divides the quotient; and dividing the divisor multiplies the quotient (Art. 83, c and d).

3d. Multiplying both dividend and divisor by the same number; or dividing both by the same number does not affect the quotient (Art. 84, a and b).

EXAMPLES IN THE FOREGOING PRINCIPLES.

1. How many bushels of corn at $1 per bushel must be given for 6 barrels of flour at $7 per barrel?

2. How many barrels of apples at $2 per barrel must be given for 8 cords of wood at $6 per cord?

3. A speculator bought 640 acres of land at $3 per acre, and sold the whole for $3200; how much did he gain by the transactions? How much per acre?

4. Bought 320 acres of land for $1760, and 320 acres more at $7 per acre, and sold the whole at $6 per acre; did I gain or lose? How much? Ans. Lost $160.

5. The expenses of a boy at school for a year are $126 for board, $24 for tuition, $15 for books, $35 for clothes, $10 for railroad and coach fare, and $9 for other purposes; what will be the expenses of 250 boys at the same rate?

6. If 3 men build 24 rods of wall in 4 days, in how many days will 5 men build 70 rods?

Ans. 7. 7. The product of 4 factors is 1155; three of the factors are 3, 5, and 7; what is the fourth?

Ans. 11.

8. How many miles per hour must a ship sail to cross the Atlantic, 2880 miles, in 12 days of 24 hours each?

9. The first of 3 numbers is 6, the second is 5 times the first, and the third is 4 times the sum of the other two; what is the difference between the first and third?

85. A more brief statement of these principles: First? Second? Third?

10. Sold two cows at $30 apiece, 3 tons of hay at $20 per ton, 50 bushels of corn for $50, and 10 cords of wood at $7 per cord, and received in payment $200 in money, a plow worth $15, 50 pounds of sugar worth $5, and the balance in broadcloth at $4 yer yard; how many yards did I receive? Ans. 5.

11. In how many days of 24 hours each will a ship cross the Atlantic, 2880 miles, if she sails 10 miles per hour?

12. If I receive $60 and spend $40, per month, in how many years of 12 months each shall I save $2160? Ans. 9.

13. What is the value of 27 hogsheads of molasses at $32 per hogshead?

14. What is the value of 87 yards of cloth at $4 per yard?

15. Bought 87 acres of land at $50 per acre, and paid $3150 in cash, and the balance in labor at $240 a year; how many years of labor did it take?

16. Bought 42 yards of cloth at 15 cents per yard, and paid for it in corn at 90 cents per bushel; how many bushels did it take?

17. If I take 13729 from the sum of 8762 and 14967, divide the remainder by 50, and multiply the quotient by 19, what is the product? Ans. 3800.

REDUCTION.

86. All numbers are simple or compound.

A SIMPLE NUMBER consists of but one kind or denomination; as 2, $4, 8 books, 5 men, 6 days, 10 miles.

A COMPOUND NUMBER is composed of two or more denominations; as 4 days and 7 hours; 3 bushels, 2 pecks, and 5 quarts; 5 rods, 4 feet, and 6 inches.

All abstract numbers (Art. 2) are simple.

86. What is a Simple Number? A Compound Number? An Abstrac Number, is it simple or compound?